Class 08

Campuswire Discussion Post

• Read Section 2.2.5 in your OpenIntro Statistics text book

• Read this opinion piece on the use of pie charts to visualize proportions

• Answer the question on Campuswire for additional participation points

• It will not be available after Friday 9/20/24

Example: Bar Plot of the Count of Students by Reputation Level

Example: Histogram of Reputation Points

Example: Rain Cloud Plot of Reputation Points

Example: Boxplot of Median vs Mean in Reputation Points

Homework 1 Learnings (part II)

You have done basic statistical analysis.

• You loaded data into a project’s environment in RStudio

• You inspected that data to determine the data type of the variables

• You created a codebook describing the variable

• You analyzed the relationship between 2 numerical variables

• You communicated your findings about that relationship

Homework 2 Objectives

• Effectively describe numerical distributions

• Select the appropriate summary statistics based on distribution shape

• Match numerical distributions to their summary statistics

• Calculate proportions from a contingency table

Late Policy

“This homework is due by 6:00pm on Friday, 9/20/24. No credit will be lost for assignments received by 7:00pm to account for issues with uploading. 10% of the points will be deducted from assignments received by 9:00am on Saturday, 9/21/24. Assignments turned in after this point are only eligible for 50% credit, so it benefits you to turn in whatever you have completed by the due date.”

Last Time…

• Contingency tables: counts and proportions (frequencies)

• Visualizing frequencies: bar plots, mosaic plots

• Describing numerical relationships: linear vs nonlinear, strong vs weak

• Visualizing 3+ variables

Today…

• Review Survey 1 Results

• Sarah’s Objectives

• Introduce Chapter 3 on Probability

Survey 1 Results

Survey 1: Bar Plot of Surveys by Date

Survey 1: Contingency Table of Graduation Year by Personal Pronouns (Counts)

Survey 1: Contingency Table of Graduation Year and Personal Pronouns (Proportions by Row)

Survey 1: Contingency Table of Graduation Year by Personal Pronouns (Proportions by Column)

Survey 1: Majors

Survey 1: Bar Plot of Motivations for Class

Survey 1: Bar Plot of What you Hope to Learn

Survey 1: Contingency Table of Level of Excitement and Coding Experience (Counts)

Survey 1: Contingency Table of Level of Excitement and Coding Experience (With Binning)

Survey 1: Mosaic Plot of Level of Excitement and Coding Experience

Survey 1: Bar Plot of Data Analysis Software Experience

Survey 1: Bar Plot of Concerns about Class

Survey 1: Bar Plot of Favorite Class Features

Survey 1: Bar Plot of How I Can Help

Survey 1: Bar Plot of “Anything Else?” Answers

Sarah’s Objectives

• Be available. I will check in with our next survey about office hour times.

• Provide real world examples and practice problems.

• Help with coding and make how-to videos.

• Be patient and give thorough explanations.

Sarah’s Other Objectives

• Be a good teacher. Unclear how.

• Get students to like statistics. Trying!

• Be fun, helpful, passionate, and engaging.

• Promote good vibes. I need your help!

Your Objectives

Many people remarked that their favorite class was their favorite because of the people in it, who were engaged, had fun, and participated.

• Come prepared

• Ask and answer questions in class

• Ask and answer questions on Campuswire

• Don’t be afraid to make mistakes

Chapter 3 Objectives

• Define probability, random processes, and the law of large numbers

• Describe the sample space for disjoint and non-disjoint outcomes

• Calculate probabilities using the General Addition and Multiplication Rules

• Create a probability distribution for disjoint outcomes

Defining Probability

What does the word probability mean to you?

People interpret probability differently

Did your estimate fall within these ranges? Are these ranges reasonable?

So what is probability?

• A random process is one where you know which outcomes are possible (i.e. the sample space) but you don’t know which outcome comes next

• Examples of a random process: coin toss, die roll, stock market

How do you know a process is random?

Both Apple and Spotify took steps to make their “shuffle” features less random after complaints from users.

A brief history of “Shuffle”

• January 11, 2005 – Apple releases the iPod Shuffle, a small device capable only of playing music randomly

  • September 7, 2005 – Apple offers “Smart Shuffle” in response to complaints, which controlled how likely songs from the same album or artist would play close together

• July 2011 – Spotify launches in the United States using the Fisher-Yates Algorithm, which is like picking tickets out of a hat until no more remain

  • February 2014 – Spotify modifies their sampling algorithm to ensure an even distribution across albums/artists

What went wrong?

• The human brain is good at finding patterns in noise, even when there are none

• If an artist is repeated “too soon”, the listener doesn’t feel the order is random

• We perceive a “random” distribution as also being “uniform” and “fair”

So why didn’t we like a “true” random shuffle?

• Songs not evenly distributed across albums and artists on a playlist

  • Some albums/artists may play more often than others because they have more songs

  • Artists/albums with more songs also more likely to play sequentially

• A true random shuffle might play the same artist multiple times in a row

  • It’s unusual but not impossible to roll a 1 on a die 6 times in a row

Example: Spotify Playlists

What if shuffle was truly random?

Each time the song changes, every song on the playlist is eligible to be played next

• Does not matter if the song was just played

• Does not matter who the artist is

We call this sampling with replacement.

• Like drawing a playing card, looking at it, then putting it back in the deck before the next draw.

• Repetition of outcomes is possible.

How often were artists repeated during Spotify’s original shuffle?

What is the probability of hearing a song by Chappell Roan?

• There is our “observed” probability that the next song is by Chappell Roan

  • Sample proportion (\(\hat{p}_n\))

• There is some “true” real-world probability that the next song is by Chappell Roan

  • Population proportion (\(p\))

Defining the sample space

The sample space is the total collection of possible outcomes for a random process.

• Die rolls: 1, 2, 3, 4, 5, 6

• Coin flips: heads, tails

• Stock market: up, down, no change

Here, the sample space is the songs on the playlist (n = 50) and the artists who perform them (n = 26).

Calculating probabilities

• Probabilities are proportions, or the number of observations with a particular value divided by the total number of observations (\(n\)).

• Proportions range from 0 (no observations in data) to 1 (all observations in data)

• Also may be a percentage, ranging from 0% to 100% (multiply proportion by 100)

Proportion of Songs by Artist

Proportion of Chappell Roan songs

How well does the sample proportion represent the population proportion?

Should we listen to 1 song?

How well does the sample proportion represent the population proportion?

Should we listen to 5 songs?

How well does the sample proportion represent the population proportion?

Should we listen to 10 songs?

How well does the sample proportion represent the population proportion?

Should we listen to 100 songs?

How well does the sample proportion represent the population proportion?

Should we listen to 200 songs?

Law of Large Numbers

As more observations are collected, the sample proportion \(\hat{p}_n\) of a particular outcome approaches the population proportion \(p\) of that outcome.

Disjoint Outcomes

Outcomes are disjoint or mutually exclusive if they cannot both happen at the same time

• The next song played cannot be by both Taylor Swift and Chappell Roan

Non-disjointoutcomes can occur at the same time.

• The next song played could be by Charlie xcx OR Billie Eilish, because they collaborated on a song